Classification-based material decomposition method for photon counting-based spectral radiography: Application to plastic sorting

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Classification-based material decomposition method for

photon counting-based spectral radiography:

Application to plastic sorting

Ting Su, Valérie Kaftandjiana, Philippe Duvauchelle, Yue-Min Zhu

To cite this version:

Classification-based material decomposition method for

photon counting-based spectral radiography:

application to plastic sorting

Ting Sua,1_{, Valérie Kaftandjian}a,_{∗, Philippe Duvauchelle}a_{, Yuemin Zhu}b a_{Univ Lyon, INSA Lyon, Laboratoire Vibrations Acoustique, F-69621 Villeurbanne, France}

b_{Univ Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS, Inserm, CREATIS}

UMR 5220, U1206, Lyon, France

Abstract

Dual-energy X-ray radiography has been commonly used for materials separa-tion. However, its performance is limited, especially for the separation of close materials or the identification of multiple materials. To cope with this problem, we propose to investigate the material decomposition ability of spectral radiog-raphy based on photon-counting detector. The latter having energy resolving ca-pability can provide spectral information of several energy bins and thus enables selective imaging of multiple materials. In this framework, a classification-based patchwise regularized decomposition method was proposed to gain better dif-ferentiation between materials. It consists of performing several decompositions with reduced number of materials in the basis and classifying these decomposi-tions using their cost function values. The results on simuladecomposi-tions showed that, in the presence of Poisson noise, the method without classification can sepa-rate acrylonitrile-butadiene-styrene (ABS) from three kinds of flame retardants (FRs: brominated FR, chlorinated FR and phosphorus FR), but that the type of FR cannot be identified. With the classification technique, ABS and three kinds of FRs can be both separated and identified at the same time when the

∗_{Corresponding author}

Email address: valerie.kaftandjian@insa-lyon.fr (Valérie Kaftandjian)

thickness was as large as 2 mm or 4 mm. The results on real data from phys-ical photon-counting detector further confirm that the ABS, Br and Cl can be separated from each other.

Keywords: X-ray, radiography, photon-counting detector, plastic sorting

1. Introduction

X-ray radiography is an imaging technique that allows to view the internal structure of an object. Since the transmitted image is a 2-D projection of the 3-D object, useful information can be hidden because of physical overlapping. Dual-energy radiography using either different kVps or dual-layer detector is 5

able to obtain two radiographs or images, one at higher energy and the other at lower energy. By making weighted subtraction of these two images [1, 2] or by performing a two-material decomposition [3], elimination of overlaying materials and enhancement of the selective target material can be achieved. Dual-energy imaging has broad applications, such as rib suppression in chest 10

radiography in medical domain [4, 5], threat detection in security inspections [6, 7, 8] and waste sorting in industrial applications [9, 10, 11]. However, its material discrimination ability is limited and materials with close atomic number (Z) can barely be separated.

Thanks to the recent development of X-ray detector technology, a variety of 15

pixelated photon-counting detectors (PCD) with imaging capability have been developped [12, 13, 14, 15]. By setting different thresholds (energy bins), each pixel of PCD is capable to either discriminate the transmitted photons corre-sponding to the selected energy bins, or firstly count the photons above the thresholds and then make subtraction to obtain the counts of each energy bin. 20

tion coefficients in spectral CT were decomposed into photoelectric absorption, Compton scattering and K-edge components, showing that dual contrasts can be well separated and quantified. Such K-edge technique is confined to the dis-crimination of high-Z materials with their K-edge inside the detection energy range. Other works [23, 24, 25, 26] considered the attenuation as a combina-30

tion of several basis materials, whose distribution can therefore be quantified through decomposition process. Such approach is particularly suitable for the discrimination of lighter materials that have much lower K-edge energy beyond the detection energy range. In our previous work [27, 28], a patchwise regular-ized decomposition method based on basis materials combination was proposed 35

for iron determination; the method allows iron to be quantitatively separated from calcium, potassium and water. However, potassium and calcium cannot be discriminated from each other since they have close Z numbers and hence close attenuation properties. The above material decomposition methods ini-tially proposed for CT can be applied to performing radiography, thus resulting 40

in the so-called PCD-based spectral radiography named as spectral radiography in the following for simplicity. In the present work, we propose a new material decomposition method for spectral radiography and investigate its feasibility for plastic sorting.

Waste electrical and electronic equipment (WEEE) has been increasing rapidly 45

due to the development of electronic industry. In European Union, the amount of WEEE generated in 2005 is 9 million tons and this number is supposed to grow to 12 million by 2020 [29]. WEEE covers a wide variety of products such as lamps, hairdryers, computers, TV-sets, fridges and cell phones. This kind of waste contains various compositions among which polymers play an important 50

part [30]. Acrylonitrile-butadiene-styrene (ABS) is a major component among all the polymers. Due to the presence of electronic power of EEE, the plastic materials should meet high fire safety standards. However, this is not possible to realize with pure polymers, therefore flame retardant (FR) are added to change flammability of plastics and increase the fire resistance [31]. Halogenated or-55

and alicyclic compounds) and phosphorus compounds (e.g. organophosphates, halophosphates, phosphine oxides and red phosphorus) are two important FRs [32]. Recycling of plastics from WEEE is challenging because of the existence of FRs (especially the brominated and chlorinated FRs), which might result in 60

serious environmental pollution. Therefore, sorting technique is important for subsequent treatment and recycling of WEEE to avoid major environmental and health problems. As mentioned before, some researches using dual-energy radiography for waste management have been reported. For example, Montag-ner et al. [9] showed that heavy FR containing bromine (Br) is easier to be 65

detected, but that chlorinated (Cl) FR can be sorted in the same family as Br, while the lighter one containing phosphor (P) remains undetectable when material thickness is less than 10 mm. To overcome these limitations, PCD-based spectral radiography appears an interesting solution due to its ability of decomposing multiple components of an object with a single scanning.

70

The purpose of the present work is to investigate the method of material decomposition for spectral radiography to discriminate plastic material from several FRs and further identify the type of FR. Based on our preliminary results reported in the ECNDT 2018 conference [33], the present paper aims to propose a new material decomposition method, which consists of performing several 75

decompositions with reduced number of materials in the basis and classifying these decompositions using their cost function values. Both simulation and experimental results will be demonstrated.

This paper reuses a substantial part from one of the co-author’s own thesis [28] with permission. More precisely, the reused contents appear mainly in 80

sections I to IV and in Appendix.

2. Classification-based decomposition method

method is to distinguish materials with close properties, and the performance of decomposition decreases when increasing the number of materials in the basis. To cope with the difficulty, the idea here is to perform several decompositions with reduced number of materials in the basis and classify these decompositions using their cost function values, thus leading to the so-called classification-based 90 PRD (CPRD) method. Acquisition patch C Decomposition Group 1 (fval1) Decomposition Group g (fvalg) fvalg<Others? Adopt gth decomposition; Absent materials = 0 End Y Decomposition Group G (fvalG)

...

...

Figure 1: Flowchart of the decomposition method with classification for one patch. fvalg represents the cost function value obtained by the gth decomposition with basis materials of

Group g.

CPRD method also works in the way of patch-by-patch decomposition, but for several groups of materials. Figure 1 illustrates the procedure of this method for one patch. The principle is to firstly choose several groups of basis materials ( Group 1, Group 2, ... Group g, ..., Group G), in which the same material can 95

belong to several groups. Then, for each patch, we perform G times decomposi-tion by carrying out PRD on each of basis material groups. For a Group g, the output is a set of decomposed density integrals at each pixel of the patch and a unique cost function value. After this step, we select the basis material group having the smallest cost function and we adopt the set of concentration values 100

concentration of other materials inside this patch are set to 0. Finally, by re-peating the above procedure for all the patches, we obtain the final image of each material.

More precisely, we consider the attenuation µ(−→x , E) as a combination of

105

M basis materials, where −→x represents location and E represents energy. The

basis materials can be chosen according to the prior knowledge of the scanned objects. We then have:

µ(−→x , E) =

M

∑

α=1

ρα(−→x )µmα(E), (1)

where ρα(−→x ) denotes the density of material α at point −→x and µmα(E) the mass

attenuation coefficient of material α at energy E. According to Beer-Lambert 110

law, the expected number of photons λi, in energy bin Bi(i = 1, 2...N, N is the

total number of energy bins) can be expressed as:

λi = E∑f(i) E=Es(i) D(E)N0(E)exp[− ∫ µ(−→x , E) ds] = Ef_∑(i) E=Es(i) D(E)N0(E)exp[− ∫ ∑M α=1 ρα(−→x )µmα(E) ds] = E∑f(i) E=Es(i) D(E)N0(E)exp[− M ∑ α=1 µmα(E)Pα(sx, sy)], (2) with Pα(sx, sy) = ∫ ρα(−→x ) ds, (3)

where Es(i) and Ef(i) denote respectively the start and final energies of bin Bi,

D(E) is the detector absorption efficiency, N0(E) is the number of photons in the initial spectrum at energy E, and Pα(sx, sy) is the density integral that is

115

line integral of object density ρα(−→x ) along the measured projection path, with

(sx, sy) representing the index of detector pixels.

To estimate Pα(sx, sy), we build an objective function that combines the

log-least squares [34, 35], and a regularization term R(PC

α) for each small patch

C on the attenuation image to reduce the effect of noise and enforce smoothness:

PαC(sx, sy) = arg min PC α(sx,sy) { ∑ (sx,sy)_∈C N ∑ i=1 [ln(λi(PαC)) − ln(mC i )] 2_{+ rR(P}C α)}, (4) where mC

i is the measured number of photons in energy bin Bi and within

patch C. r denotes the relaxation parameter. R(PC

α) is the sum of the L2

regularizations of gradient images of PC α: R(P_αC) = M ∑ α=1 || ▽ PC α|| 2 2 = M ∑ α=1 ∑ (sx,sy)∈C (sx−1,sy)∈C (sx,sy−1)∈C {[PC α(sx, sy)− P C α(sx− 1, sy)] 2 +[P_αC(sx, sy)− P_αC(sx, sy− 1)]2}. (5)

3. Description of the spectral radiography simulation

We use the Virtual X-ray Imaging (VXI) software [36], which was developed 125

Table 1: Components of three ABS-FR materials used for the phantom.

Material(ABS-FR) ABS- ABS-

ABS-TBBPA DDC-CO RDP

Density of material ρ(mixture) 1060 mg/cm3 _{1060 mg/cm}3 _{1060 mg/cm}3 Chemical formula of FR C15H12Br4O2 C18H12Cl12 C30H24O8P2

Mass % (ω)of FR 15% 15% 15%

Mass % (ω) of Br, Cl and P, respectively 8.82% 9.76% 1.62%

ρeff*of Br, Cl and P, respectively 93.5 mg/cm3 103.5 mg/cm3 17.2 mg/cm3 *_{: ρ}

eff is the effective density calculated by ρeff(α) = ρ(mixture)× ω(α). For example, ρeff(Br) = ρ(mixture)× ω(Br) = 1060 × 8.82% = 93.5 mg/cm3_;

ρeff(ABS) = ρ(mixture)× ω(ABS) = 1060 × (1 − 15%) = 901 mg/cm3.

3.1. ABS-FRs phantom ABS- TBBPA (Br) ABS- DDC-CO (Cl) ABS- RDP (P) ABS Thickness d=0.3mm d=0.5mm d=2mm d=1mm d=4mm 10 mm 10 m m

Figure 2: Illustration of the ABS-FR phantom used for spectral radiography imaging. Mate-rials in the cubes are given in Table 1.

We simulate a phantom containing multiple polymer+FR mixtures. As men-130

and electronic equipments. Therefore, we selected it as the polymer material for investigation. We also chose three kinds of commonly used flame retardants, including the brominated, chlorinated and phosphorus FRs. They are respec-tively tetrabromobisphenol A (TBBPA), dechlorane plus (DDC-CO) and resor-135

cinol bis(diphenyl phosphate) (RDP). Three ABS-FR materials were obtained by mixing each FR with ABS at mass percentage of 15%. Detailed information of these materials is summarized in Table 1.

The simulated phantom is composed of multiple cubes with height of 10 mm, width of 10 mm and different thicknesses. As shown in Figure 2, each column of 140

cubes are of the same material (denoted on the top) and each row of cubes are of the same thickness (denoted on the left). Figure 3 plots the mass attenuation coefficients of the components contained in the phantom: ABS, Br, Cl and P, which will be used to distinguish different ABS-FR materials.

30 40 50 60 70 80 Energy (kev) 10-1 100 101 102 m (cm 2/g) Br Cl P ABS

Figure 3: Mass attenuation coefficients (µm) of components (ABS, Br, Cl and P) within the detection energy range from 30 keV to 90 keV. µmof ABS is calculated according to formula

µm=∑αωαµmα, with ωα and µmα representing the weight fraction and mass attenuation coefficient of each element α of the compound. Data taken from [37].

From the decomposition method described in the above, the finally obtained 145

Pαfor material α is the line integral of material density ρα(Equation 3). In this

Table 2: Theoretical values of Pαfor each material. Material ABS has two columns: the first is ABS pure, which corresponds to the material of the 4th column of cubes in the phantom shown in Figure 2, and ABS in mixture corresponds to the ABS compound in the other cubes of Figure 2. Thickness (mm) ABS pure (mg/cm2₎ ABS in mixtures (mg/cm2₎ Br in ABS-TBBPA (mg/cm2₎ Cl in ABS-DDC-CO (mg/cm2₎ P in ABS-RDP (mg/cm2₎ 0.3 31.80 27.03 2.81 3.11 0.52 0.5 53.00 45.05 4.68 5.18 0.86 1 106.00 90.10 9.35 10.35 1.72 2 212.00 180.20 18.70 20.70 3.44 4 424.00 360.40 37.40 41.40 6.88

(sx, sy) of the transmitted image is given by

Pα(sx, sy) = d(sx, sy)× ρα(sx, sy), (6)

where ρα equals the effective density calculated in Table 1. Table 2 lists the

theoretical values of Pαfor materials ABS, Br, Cl and P at various thicknesses.

150

3.2. Acquisition parameters and system geometry

Planar ar ray P CD DDC=3 mm DSC=2000 mm X-ray source

The simulated system used a point X-ray source of tungsten target material and the target angle was 17◦. According to reference [38], tube voltage was set to be 100 kVp, tube current 15 mA and exposure time 1 s. The X-ray spectrum was computed based on Birch & Marshall model [39] without filtration. A 155

90×112 CdTe detector array with pixel size of 0.5 mm × 0.5 mm and thickness

of 3 mm was simulated. Six energy bins were set to be evenly distributed from 30 keV to 90 keV. We assumed that the detector has ideal energy resolution, detector absorption efficiency was simulated as the fraction of photons absorbed by the detector material. The distance from X-ray source to phantom center 160

was 2000 mm and the distance of detector to phantom center was 3 mm. Figure 4 is a schematic view of the simulated spectral radiography system. Photonic noises were simulated by simple Poisson distribution with means equaling the expected numbers of photons acquired for each energy bin [40, 41]. This process was implemented in Matlab using command poissrnd.

165

4. Simulation Results

We have simulated two sets of spectral radiography acquisition data: with and without photon noise. To better evaluate the performance of the proposed CPRD method, it was also compared with PRD method without classification.

4.1. Decomposition results of PRD method

170

Regarding the components of the phantom, it is normal to consider a four basis material decomposition: ABS, Br, Cl and P. However, our experiment results show that P and Cl can barely be separated from each other since they have too close atomic number and hence close attenuation properties. Therefore, we decompose the radiographic images into three basis images corresponding to 175

ABS, Br and Cl and expect that P will also be present in the Cl basis image. The patch size is set to be 2×2 pixels.

400 300 200 100 400 300 200 100 400 300 200 100 30 20 10 0 30 20 10 0 30 20 10 0 40 30 20 10 0 40 30 20 10 0 40 30 20 10 0 r=2.51·104_.

- In the ABS basis image, all cubes are visible as expected, because ABS is present in both mixtures and pure ABS.

- In the Br basis image, the ABS-TBBPA cubes are well separated and highlighted with the Pα close to the true values, and no other FRs are

present. 185

- In the Cl basis image, all the ABS-DDC-CO cubes are well highlighted as expected. At the same time, ABS-RDP is identified as Cl as expected, with Pαclose to true values. However, for small thicknesses (d≤ 2 mm),

a slight amount of ABS-TBBPA is misidentified as Cl.

Thus in the noise-free situation, ABS and Br can be well separated and 190

identified. The determination performance of other FRs improves with the increase of object thickness: when d = 4 mm, the FRs containing Cl and P are present in the Cl basis image without cross-talk with the ABS-TBBPA material, one can further distinguish the two FRs according to their large density difference. In other words, if the thickness is known, the density can be used to 195

identify the right material.

When Poisson noise is added, we need to reconsider the relaxation parameter

r. r is determined as 2.51·104_{in this situation according to the L-curve criterion.} Some details of r selection using the L-curve method are introduced in the Appendix The decomposition results when r = 0 and r = 2.51· 104 _{are shown} 200

in Figure 5(b) and (c). It can be seen that when r = 0 (without regularization), the ABS-TBBPA cubes and ABS-DDC-CO cubes appear in both Br basis image and Cl basis image, making them impossible to be distinguished. Moreover, the decomposed images suffer from heavy noise. When the relaxation parameter was set to r = 2.51· 104_{, the obtained images are less noisy than those in the former} 205

4.2. Decomposition results with the new CPRD method

Instead of the 3-material (ABS, Br and Cl) decomposition using PRD method, we perform two independent decompositions with two groups of basis materi-210

als, group 1 is ABS + Br, and group 2 ABS + Cl. We give up a third group of ABS + P because it generates comparable cost function values with group 2, resulting in severe cross-talk between Cl and P. With the selected two groups of decomposition, by comparing the cost function values (fval1 and fval2) at the end of each decomposition, one of the two decompositions is chosen for a 215

given patch. For example, if fval1<fval2, we choose the results of the decompo-sition that decomposes the data into ABS and Br, consequently, the values of Cl in this patch will be set to 0. After all patches being considered, full images of ABS, Br and Cl are obtained. Figure 6 shows the decomposition results of CPRD method and a reference method in the same condition as that in the 220

0 100 200 300 400 0 10 20 30 0 10 20 30 40 Br basis ABS pure ABS-TBBPA (Br) ABS-DDC-CO (Cl) ABS in mixtures ABS basis 2mm d= 0.3mm 0.5mm 1mm Cl basis 4mm (b) CPRD. With noise; r=0. (c) CPRD. With noise; r=10 4.4. (a) CPRD. No noise; r=0. (d) Reference method. With noise; ABS-RDP (P)

In the noise-free condition (Figure 6(a)), FRs containing ABS and Br are identified respectively by CPRD method in the ABS and Br basis images as 225

expected. Meanwhile, only the cubes of FR materials containing Cl and P appear in the Cl basis image, but they have significant density difference for cubes with the same thickness, therefore, they can be easily distinguished by the observer even if the concentration of FRs or thickness of cubes changes more or less.

230

In the presence of Poisson noise, Figure 6(b) shows the decomposition re-sults of CPRD method when r = 0 (without regularization). Noise can be observed in decomposition images, however, material separation is better than with PRD method (compared with Figure 5): only few pixels containing Cl are misidentified in the Br basis image and vice versa. The cross-talk situation 235

improves with the increase of thickness. In Figure 6(c), when r = 2.51· 104 (with regularization), the material separation is further enhance to some extent compared with Figure 6(b): less pixels of ABS-DDC-CO cubes appear in the Br basis image, and only the ABS-DDC-CO and ABS-RDP cubes are highlighted in the Cl image when d = 2 mm or 4 mm, like the results in noise-free case. 240

In contrast, the reference method fails to separate Br and Cl. As can be seen in 6(d), the cubes containing Br also appear in the Cl basis image, no matter what thickness the sample has. Therefore, the CPRD method outperforms the reference method for material decomposition.

A more quantitative comparison is done in Figure 7, where the profiles of 245

computed Pα with PRD and CPRD method are plotted together with the

the-oretical values (along dash lines in both Figure 5 and 6). It is observed that the CPRD method yields closer results to true values with respect to the PRD method especially for the Cl basis curve, where ABS-TBBPA cubes are mistaken as Cl containing FR by PRD method. It is noteworthy that the theoretical val-250

ues of Pαfor the ABS-RDP cube in the Cl basis image are set to 1.72mg/cm2

(corresponding to Table 2, the Pα of P in ABS-RDP with thickness of 1 mm)

sorting application, no matter when Poisson noise is or is not present. ABS 255

and Br containing FR can be identified in their corresponding basis images. Furthermore, when the cube thickness is increased to 2 mm or 4 mm, the de-composition results of Poisson noise condition become much closer to those of noise-free condition, where ABS and the FRs containing Br, Cl and P can be identified simultaneously. 260 Column 1 ABS-TBBPA (Br) Column 2 ABS-DDC-CO (Cl) Column 3 ABS-RDP (P) Column 4 ABS pure

Figure 7: Performance comparison of PRD and CPRD methods in noise-free condition: 1-D profiles along the dash lines in Figures 5 and 6. ABS basis (top), Br basis (middle) and Cl basis (bottom). Black curves with point marker represent the theoretical density integrals of basis materials (Pα for thickness of 1 mm in Table 2)for perfect decomposition. Blue curves with triangle marker represent the calculated values using PRD method and the magenta curves with cross marker using CPRD method.

5. Preliminary experimental results

detector from Detection Technology company. The detector has linear array structure with 256 detector pixels, each pixel is of 0.8 mm pitch size. The 265

detector provides the possibility of setting 64 uniformly distributed energy bins between 20 keV and 160 keV. Considering our tube voltage of 110kVp for this experiment, we used data of the first 36 bins (20 keV to 110 keV) to perform material decomposition. As shown in Figure 8, the object being scanned is composed of 5 materials: polyvinyl chloride (PVC), ABS with 20% Br, ABS 270

with 5% Cl, polypropylene (PP) with 20% CaCO3, and ABS pure. Acquisition parameters were set as: tube current 200 µA, exposure time 100 ms, source to object distance 46 cm and object to detector distance 13 cm.

PVC ABS pure ABS+5%Cl PP+20%CaCO3 ABS+20%Br Specimen

Figure 8: Illustration of experimental specimen (left) and system geometry (right).

Because of the linear structure of the detector, we have acquired single line data of 256 pixels. Therefore, the patchwise regularization, which requires 2 275

Figure 9: Decomposition results of ABS (a), Br (b), Cl (c) and Ca (d). The green check mark indicates a good decomposition while the red cross mark indicates a false decomposition.

The decomposition results are shown in Figure 9. It can be observed that 280

all 5 materials are visible in Figure 9(a), since they either really contain ABS or contain components like PVC or PP that have similar attenuation properties to ABS. The pure ABS curve appears only in Figure 9(a), which indicates a good separation from other basis materials. In Figure 9(b), component Br is well discriminated. In Figure 9(c) the Cl component in PVC is well identified. 285

However, a mistaken peak of PP with 20% CaCO3also appears, indicating that Ca probably cannot be separated from Cl. Also, the sample containing 5% Cl remains undetected.

In short, the proposed method achieves to identify ABS, Br as well as Cl with relatively higher concentration, but materials with close atom numbers, 290

such as Ca and Cl remain unseparated.

6. Discussions

We have proposed a classification-based material decomposition method, which achieves to identify several different plastic materials in both simulation and experimental validations. There are several parameters and factors that 295

may influence the decomposition performance as discussed below.

Choice of basis materials. During simulation, we have found that using

have also performed a 3-group decomposition with basis materials like, Group 1 (ABS + TBBPA), Group 2 (ABS + DDC-CO) and Group 3 (ABS + RDP). The results showed that with cube thickness of 2 mm and 4 mm, ABS and three FRs can be identified according to their corresponding decomposition image. That is to say, DDC-CO and RDP can be distinguished from each other directly by 305

decomposing them into their corresponding basis, instead of by their density difference in the same basis (Cl) image, as described in the above results of the CPRD method. We have not chosen these groups of basis materials with better performance because they are too specific. However, for the application where the components of object are well known, appropriate choice of basis materials 310

can yield better decomposition performance.

Patch size. In the present study, we have used a patch size of 2× 2 pixels.

To investigate the influence of patch size on decomposition performance, we plot in Figure 10 the curve of quantification error versus patch size, where quantifi-cation error represents root-mean-square-error (RMSE) of decomposed Br basis 315

image. As observed, 2× 2 patch size gives the smallest RMSE. Quantification error increases when patch size is larger than 2. This could be caused by the fact that large patch size results in over-smoothed image [43]. As a result, since there exist sharp grid-like edges throughout our entire image, over-smoothing of these edges leads to larger quantification errors.

320 0 2 4 6 8 10 12 Patch size 1 1.5 2 2.5 3 3.5 4 4.5 RMSE of Br basis

Relaxation parameter r. Although the separation between materials is

not enhanced (for PRD method) or slightly enhanced (for CPRD method) by choosing r according to L-curve method, it is still meaningful that noise (indi-cated by the variation of pixel values) of decomposition image can be decreased, which is obvious in Figure 5. We have applied L-curve method to certain re-325

gion of the radiographic images, and used the same r for all patches. However,

r should perhaps be changed for different patches if the signal-to-noise ratio

varies. Therefore, further improvement could be achieved by tuning the relax-ation parameter in each patch.

Placement of energy bins. In the present work, we have used 6 uniformly

330

distributed bins within detecting energy range to evaluate the performance of our proposed method in a general way. However, according to some researches [44, 45], appropriate selection of energy bins may have large influence on decom-position performance, especially when the total number of bins are small. In Figure 11(b), we demonstrate the decomposition results using a different strat-335

egy to place 6 energy bins. More bins were placed in lower energy on purpose since lower energy carries more material-differentiation information. We can observe performance improvement compared to Figure 11(a). From the ROI marked by the red circles, Br was better recognized with less cross-talk in Cl basis. Moreover, the calculated RMSE of Br basis image decreased from 1.23 340

0 100 200 300 400 0 10 20 30 0 10 20 30 40 0 100 200 300 400 0 10 20 30 0 10 20 30 40 Br basis Cl basis (a) (b) ABS basis

Figure 11: Comparison of decomposition results using different strategies of selecting energy bins. (a) 6 energy bins are uniformly distributed between 30keV and 90 keV, which is the same as Figure 6 (c). (b) More bins were placed in lower energy range as follows: 30 - 34, 35 - 41, 42 - 50, 51 - 61, 62 - 74 and 75 - 90 keV.

Detector response. In the present work, simulation validation only

con-siders the absorption efficiency of the detector. However, in reality, photon counting detector’s response function can be modeled by mixture of Gaussian functions allowing taking into account several artifacts such aselectronic noise, 345

charge sharing, pulse pile-up, K-escape and Compton scattering [46]. Electronic noise is an important factor that limits the energy resolving capability of de-tector, it results in spectral resolution degradation and potential false counts especially at low energy. [47]. Charge sharing effect occurs when the charge cloud generated from X-ray interaction is shared by neighboring pixels and re-350

is due to the characteristic X-rays induced by photoelectric effect. It can lead to different consequences: the recorded energy is lowered by K-shell energy, or two counts are generated, or still pile-up effect is induced [49].Compton scat-tering refers to Compton effect-induced scattered photons that are detected by an adjacent pixel or leave the PCD completely, resulting in long tail in lower 360

energy. All these effects lead to distorted detector response and reduction of energy resolution. The degraded performance of material decomposition caused by discrepancy between theoretical and measured data can be addressed by two approaches. The first one is to characterize detector response by synchrotron radiation, radioactive isotopes or x-ray fluorescence [50], and then incorporate 365

this information into material decomposition [51]. The second one is to cali-brate material’s linear attenuation coefficient by scanning samples with known components and thickness, so as to take into consideration the influence of de-tector response. The second approach is more flexible, easy to accomplish and efficient according to our preliminary test on experimental data. Though this 370

strategy was not used for the demonstrated results here, we can still observe pretty good decomposition performance, indicating that our proposed method is able to process real experimental data.

There are certain limitations of the proposed method. Firstly, the CPRD method is not enough robust to noise when the object thickness is small. Sec-375

ondly, the decomposition of the two FRs with similar properties (DDC-CO and RDP), even in the noise-free case, depends on their density difference in the Cl image. Therefore, in case where the thickness information of objects is un-known, or the two materials overlap in the X-ray direction, the decomposed density will be largely influenced, yielding the materials indistinguishable. Al-380

though an optional solution has been mentioned in the above, which consists in using directly DDC-CO and RDP as basis materials, but in a more generalized situation, close materials are still hard to be distinguished. Thirdly, the pro-posed CPRD method performs several decompositions into different groups of basis materials and then uses a classification technique to decide the right type 385

re-duces the running time of each decomposition, compared with a normal one with multiple basis materials, but on the other hand, the computation time can be in-creased because of extra decomposition procedures. For instance, in the present study, the 2-group (two basis materials in each group) CPRD decomposition 390

has about 36% longer running time than the 3-material PRD decomposition.

7. Conclusions

To investigate the ability of PCD-based spectral radiography for plastic sort-ing, we have proposed a classification-based decomposition method and evalu-ated its performance through both simulation and experimental validations. 395

The simulation results showed that the proposed CPRD method is able to dis-tinguish ABS and the FRs containing Br, Cl and P at the same time if the cube thickness is as large as 2 mm or 4 mm. Moreover, the quantified material densities agree well with theoretical values. Experimental validations further confirm that ABS, Br and Cl can be separated from each other, but that ma-400

terials with close atom numbers, i.e. Ca and Cl remain unseparated. Despite some limitations, our results suggest the possibility that PCD-based spectral radiography can be used to sort different types of plastic materials.

In addition to plastic sorting presented in the present work, the proposed method can be readily applied to other applications such as industrial and med-405

ical ones. For example, in explosive detection, the explosive substances have close X-ray attenuation properties with some common materials such as sugar and polyethylene, a spectral radiography scan followed by the CPRD decom-position may improve their separation. Furthermore, the proposed method can be adapted to spectral photon-counting CT by adding a reconstruction step to 410

transform the projection density integral information into cross-sectional den-sity distributions. Thus spectral CT inspections could benefit from the proposed method too, such as abdominal imaging and atherosclerosis imaging, where material-specific imaging is of great importance for characterizing lesions or high risk plaques.

Appendix A. Determination of r with L-curve method

The L-curve is a log-log plot of the norm of a regularized solution versus the norm of the corresponding residual, where the “corner” of the curve corresponds to the selected relaxation parameter [52]. The solution and residual norms correspond to the objective function value and R(PC

α) value in Equation 4. We

420

choose a region of interest on the radiographic image for the determination of

r, which is shown in Figure A1.

ABS- TBBPA (Br) ABS- DDC-CO (Cl) ABS- RDP (P) ABS d=0.3mm d=0.5mm d=2mm d=1mm d=4mm ROI ROI 20 pixels 20 p ix els 10×10 patches

Figure A1: Illustration of ROI for the determination of r.

The ROI contains 20×20 pixels. Given that the patch size is 2×2, there

are 10×10=100 patches within the ROI. For a given r, we define the solution

norm and residual norm as the average values calculated over the 100 patches. 425

We plot the L-curve with r varying from 10−10 to 1010_{, as shown in Figure} A2 (a). It is observed that the “corner” exists between r = 104 _{and r = 10}5_. Furthermore, by plotting the same curve for a set of r between 104_{and 10}5_{, we} have the curves shown in Figure A2(b). From this figure, we select r = 2.51·104 as the relaxation parameter for all patches.

10-5 10-4 10-3 10-2 10-1 Residual norm 10-20 10-10 100 Solution norm r =104 r =105 (a) 6 8 10 12 14 Residual norm 10-5 10-14 10-13 10-12 10-11 Solution norm r =104 r =105 r =2.51 104 (b)

Figure A2: The L-curve as a function of residual norm for different values of the relaxation parameter r (a), and a magnified view of the blue circle region, where r varies from 104 _to

105 _(b).

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